The strength of electron electron correlation in Cs3C60

Cs3C60 is an antiferromagnetic insulator that under pressure (P) becomes metallic and superconducting below Tc = 38 K. The superconducting dome present in the T − P phase diagram close to a magnetic state reminds what found in superconducting cuprates and pnictides, strongly suggesting that superconductivity is not of the conventional Bardeen-Cooper-Schrieffer (BCS) type We investigate the insulator to metal transition induced by pressure in Cs3C60 by means of infrared spectroscopy supplemented by Dynamical Mean-Field Theory calculations. The insulating compound is driven towards a metallic-like behaviour, while strong correlations survive in the investigated pressure range. The metallization process is accompanied by an enhancement of the Jahn-Teller effect. This shows that electronic correlations are crucial in determining the insulating behaviour at ambient pressure and the bad metallic nature for increasing pressure. On the other hand, the relevance of the Jahn-Teller coupling in the metallic state confirms that phonon coupling survives in the presence of strong correlations.

of Cs 3 C 60 shows the emergence of superconductivity from an insulating antiferromagnetic parent state, resembling what found in other high-temperature superconducting families [17][18][19] and calls for a deeper understanding of the role of electron-electron interactions in A 3 C 60 compounds 20,21 , and in particular in Cs 3 C 60 22 .
In this work we have combined novel infrared experimental data with theoretical calculations to shed more light on the strength of electron correlation and on the evolution of the Jahn-Teller interaction in A15 Cs 3 C 60 under pressure. Infrared spectroscopy is a powerful technique to study the low-energy electrodynamics of solids as it yields information on both the electronic and vibrational degrees of freedom. By measuring the optical conductivity, the pressure-driven metallization process can be traced by the emergence of a Drude term in the spectra. On the other hand, a splitting of the phonon lines indicates a symmetry reduction, possibly associated with a Jahn-Teller distortion.
In this paper we investigate the low-energy electrodynamics and the insulator to metal transition of Cs 3 C 60 under pressure at room temperature. We find that a strongly correlated bad metallic state is established at rather low pressures in concomitance with the Jahn-Teller distortion which maintains and stabilizes in the metallic phase. This result suggests that in the proximity of the Mott transition may enhance the electron-phonon interaction leading to the relatively high superconducting critical temperature in the A15 structure.

Results
Infrared Data. At ambient conditions the reflectivity at the sample-diamond interface R S−D is almost flat, and shows the infrared signatures of the T 1u vibrational modes at about 570 and 1370 cm −1 . For increasing values of the applied pressure (indicated in Fig. 1 by the grey arrow), R S−D raises towards low-frequency, suggesting the onset of a carrier-delocalisation process. In Fig. 1E are reported the optical conductivity curves σ 1 (ω) achieved by Kramers-Kronig transformation from R S−D data (see the Method section) up to 6000 cm −1 . Already at 4.0 and 6.0 kbar the conductivity level increases towards low frequency through a transfer of spectral weight from above to below an isosbestic point around 2000 cm −1 . The insulator to metal transition occurs around 13 kbar where a Drude term clearly shows up in the spectra and σ 1 (ω) monotonically increases for ω → 0. Noteworthy, in the metallic phase of Cs 3 C 60 the absolute value of σ 1 (ω) is that of a poor metal as also observed in Rb 3 C 60 , K 3 C 60 23 and in Na 2 CsC 60 24 . At the lowest pressure the phonon mode at 570 cm −1 is symmetric (see Fig. 1F), as found also in ref. 25. When the system at high pressure becomes metallic, this phonon mode couples with the electronic continuum and shows the signature of a Fano-like distortion, similarly to what found in the metallic A 3 C 60 (A = Rb, K) compounds. The phonon mode centered at 1370 cm −1 (Fig. 1G) shows instead a splitting in three peaks, that becomes more evident at the highest measured pressure. Let us observe that the splitting of this phonon is in excellent agreement with what shown in ref. 25, also perfectly confirming the calculations performed in ref. 26.

Discussion
Electronic correlation. To assess the importance and the strength of the Coulomb repulsion U in determining the ground state of Cs 3 C 60 we can compare the experimental results with theoretical calculations.
The ratio between the experimental kinetic energy (K exp ) and that obtained via band structure "mean-field" calculations (K LDA ) can be used to determine the "degree of correlation" of a material. Such a ratio spans between 0 (characteristic of a Mott insulator) to 1, as in the case of conventional metals [27][28][29][30] . The K exp /K LDA ratio is shown in Fig. 2 for Cs 3 C 60 . K exp was obtained from the integral of σ 1 (ω) for a cut-off frequency of 900 cm −1 , which captures pratically all the Drude spectral-weigth, as can be seen in the Drude-Lorenz fit shown in Fig. 1D. The error-bars in Fig. 2 take into account an indetermination of ± 100 cm −1 in choosing the cutoff energy for the Drude term. K LDA are instead calculated through Density-Functional Theory (DFT) using the crystal structure relative to different pressure values (see Methods). K exp /K LDA values are below 0.1, placing Cs 3 C 60 at the verge of the Mott transition 27 . With increasing pressure Kexp/KLDA raises, mirroring the increase of spectral weight at the Fermi energy, i.e. representing the emergence of a quasi-particle peak in the density of states. The Drude peak that shows up in Cs 3 C 60 above 13 kbar does not resemble that measured in the metallic phases of K 3 C 60 or Rb 3 C 60 31 .
Here, a broad Drude term can be found, superimposed to the HOMO-LUMO excitations at about 1 eV. Instead in Cs 3 C 60 , a relatively small Drude term is present, whereas a broad absorption band is still visible slightly below 2000 cm −1 .
To further understand the importance of electron-electron correlation on Cs 3 C 60 , we compare the experimental results with calculations combining DFT with Dynamical Mean-Field Theory (DMFT). The impurity model is solved at 300 K by finite-temperature exact diagonalization 32 . From DMFT we have computed K exp /K DMFT that turns out to be ( Fig. 2) nearly constant (and equal to 1), for increasing pressure. This highlights that DMFT correctly captures the pressure-driven appearance of the quasi-particle peak and its effect on the kinetic energy. (see Fig. 2). The corresponding theoretical optical conductivity curves are reported in Fig. 3 where it is possible to follow the partial closure of the gap and at the same time the growth of the Drude term by increasing pressure. The theoretical result describes the evolution of σ 1 (ω) with pressure accurately, corroborating that electron-electron correlation plays a major role in the physics of Cs 3 C 60 and mainly determines the Insulator-to-Metal transition. Noteworthy, both Scientific RepoRts | 5:15240 | DOi: 10.1038/srep15240 theoretical and experimental curves show the coexistence of the Drude term with a mid-infrared absorption band, as an infrared signature of strong correlation 33 . This also occurs at the pressure values where superconductivity is found at low temperature, confirming the belief that the superconductivity emerges from a strongly correlated metallic phase 20 .
Phonon modes. Besides the strong electron-electron interactions, the molecular Jahn-Teller effect plays a pivotal role in the physics of fullerides. The coupling with the Jahn-Teller active H 1g modes is the largest contribution to the electron-phonon coupling, and it is widely believed to be responsible of the superconducting pairing. In particular, it has been demonstrated that a Jahn-Teller coupling can survive and even benefit from the strong correlations identified in the present study 14 .
Recently some authors 25 have discussed the fingerprints of the molecular Jahn-Teller effect in the optical spectra of Cs 3 C 60 at ambient pressure. By lowering the temperature below 300 K, a splitting of this phonon mode has been detected, hinting to the presence of a dynamical Jahn-Teller distortion. While at room temperature and above, the thermal expansion results in a weaker crystal field that does not select In our study a shoulder in the 1370 cm −1 phonon is visible already at the lowest measured pressure, however the splitting becomes more evident for increasing pressure, even at room-T, where instead at The ratio of the experimental and the "mean-field" theoretical kinetic energy K exp /K LDA ratio (triangles) is compared with data from ref. 28 on V 2 O 3 and Cu. K exp /K LDA increases with increasing pressure indicating a smooth transition from a Mott insulator to a correlated metal. On the other hand, K exp /K DMFT (squares) is nearly constant at 1. This indicates that DMFT is taking correctly into account the electron-electron correlation, capturing the pressure-driven appearance of the quasi-particle peak.   ambient pressure, it was argued 25 that no preferred direction was chosen by the distorted molecule. A preferred direction seems to be present instead at high-pressure, likely due to a modification of the local crystal field with pressure. A similar behaviour has been recently discussed on Rb x Cs 3-x C 60 , where the interfulleride distance is controlled via chemical substitution rather than by external pressure 34 . Therefore, our data show that pressure can drive the system metallic while maintaining (and stabilizing) a dynamical Jahn-Teller distortion. These results confirm that the Jahn-Teller effect is not a main force in the stabilization of the insulating behavior, which is indeed solely due to the Mott localization of carriers. Interestingly, the present data show that the Jahn-Teller coupling is more effective when the system becomes metallic, either by increasing the pressure or reducing the temperature, a result fully compatible with the scenario in which a Jahn-Teller coupling leads to superconductivity in the strongly correlated metal close to the Mott transition 20 .

Conclusions
In conclusion, pressures in the kbar range are sufficient to drive Cs 3 C 60 in a metallic state, while signatures of strong electron correlation persist up to the highest measured pressure. Cs 3 C 60 is thus found to be at the verge of a Mott-Hubbard Insulator-to-Metal transition: by applying an external pressure a carrier-delocalization is induced corresponding to a poor-metallic state where dynamical Jahn-Teller distortions are maintained and stabilized.
The comparisons with DMFT theoretical calculations support the claim that electron-electron correlation is the sole responsible for the insulating state at ambient conditions and can alone explain the low-energy Cs 3 C 60 evolution across the IMT. Noteworthy, strong correlation persists at high pressure and therefore it has to be considered in order to reach a full understanding of superconductivity in Cs 3 C 60 . On the other hand, the relevance of the Jahn-Teller coupling in the metallic state confirms the theoretical scenario in which the phonon coupling survives in the presence of strong correlations.

Methods
Sample preparation and characterization. A15-rich Cs 3 C 60 sample was obtained via a solvent-mediated synthesis route. Stoichiometric amount of Cs metal (Aldrich, > 99.5% purity) and C 60 powder (MER > 99.9%) were put in a Pyrex vial in a controlled atmosphere (Ar glove-box, < 0.1 ppm O 2 and H 2 0), having particular care to avoid the direct contact between the reagents in this stage. The vial was then evacuated (P < 10 4 mbar) and placed in a methanol bath at − 60 degrees Celsius and previously degassed anhydrous methylamine (Aldrich > 98% purity) was condensed under continuous stirring. At this step the solution became dark-red, due to the dissolution of the alkali metal in the solvent. Hence, the vessel was sealed and slowly heated at T = 50 degrees Celsius and let react for one night under stirring. After reaction took place the color of the suspension became dark brown. Methylamine was slowly evaporated at − 5 degrees Celsius and the dry product was collected in glove box, then it was pelletized and further treated in dynamic vacuum for 20 h at 200 degrees Celsius. Quantitative Phase Analysis (QPA) of the product was performed by synchrotron radiation powder diffraction, and indicated a phase fraction of respectively 74(1)% A15, 14(1)% fcc and 12(1)% bco.
Infrared Spectroscopy under pressure. High pressure measurements were performed with a screw-driven opposing plate Diamond-Anvil-Cell. An Al gasket was chosen to span over the desired pressure range (0-20 kbar) in a reproducible and cyclic way. A hole of about 300 μm diameter was drilled in the pre-indented gasket. A small amount of Cs 3 C 60 sample was pressed between the diamond anvils to form a pellet with a thickness of about tens of microns 35,36 . This thickness yields zero transmission, allowing for reflectivity measurements. The pellet was then loaded in the cell with CsI as pressure medium, taking great care of obtaining a clean diamond-sample interface. As these samples are strongly air-sensitive the whole procedure was done in a glove box and the pressure cell was carefully kept closed for the experiment. The pressure was measured in-situ by the standard ruby fluorescence technique 37 with a procedure described elsewhere 38 . Infrared reflectivity measurements were performed with the aid of the Hyperion 2000 Bruker microscope coupled to an IFv66/s Michelson interferometer, exploiting the high brilliance of synchrotron radiation at the infrared beam line SISSI of ELETTRA storage ring 39 . Reflectivity at the sample diamond interface R S−D was measured with 1 cm −1 resolution between 250 ÷ 15000 cm −1 . The real part of the optical conductivity σ 1 (ω) was then extracted via Kramers-Kronig (KK) transformations taking care of the sample diamond interface 38,40 . Simultaneaous fitting of R S−D and σ 1 (ω) with a Drude-Lorenz model is performed to make sure that the KK procedure has been performed correctly.
Dynamical Mean Field Theory calculations. Our theoretical data are based on the combination of density functional theory (DFT) with Dynamical Mean-Field Theory (DMFT). The DFT band structure of Cs 3 C 60 has been calculated using the Perdew-Burke-Ernzerhof recipe for the genealized gradient approximation by means of the Quantum Espresso package 41 employing a grid of 6 × 6 × 6 k-points. The cutoff energies for wavefunctions and charge densities were set to 45 Ry and 450 Ry, respectively. The calculations are based on the A15 structure of Cs 3 C 60 with Pm3n symmetry with b.c.c. anion packing 23 at ambient pressure without performing structural relaxations, and the pressure dependence is deduced from the experimental data for the lattice spacings. A tight-binding representation of the band structure is built using Wannier90 42 to compute the maximally localized Wannier orbitals restricting to the bands originated from the t 1u LUMO.
We combine the DFT band structure with the on-site interactions acting on the three-fold degenerate t 1u manifold. We considered the large Hubbard U estimated in ref. 43, and we included an attraction term which results from the Jahn-Teller electron-phonon interaction, even if significantly reduced by the Hund's rule coupling, which involves exactly the same operators 5 . As we did not consider structural relaxations, we do not account for the static Jahn-Teller distortions, while we account for the dynamical Jahn-Teller interaction and its effects of the electronic properties. Indeed the two interaction terms have been shown to reproduce the low-temperature phase-diagram of Cs 3 C 60 within model calculations.
Here, we chose the value of the attractive interaction in order to reproduce the spectral gap of the insulating compound at ambient pressure. The value used is J = 0.07 eV. The DMFT calculations employ an exact diagonalization solver at finite temperature in the implementation of 32 4 impurity levels per orbital are included for a total number of 15 levels including the impurity site, and the full orbital rotational symmetry is implemented to reduce the Hilbert space. The optical conductivity is computed from the single-particle Green's functions including the vertex functions obtained by differentiation of the DFT band structure with respect to momentum.